98 ESSAY ON PROBABILITIES. 



and for the following reason. Prudence requires that 

 no one should expose himself to great risks of loss, and 

 does not accept it as an excuse that there was a remote 

 chance of enormous gain, or that another had the same 

 chance of the same gain or loss. Again, the mathe- 

 matical expectation is derived from the result which, as 

 can be shown, will be produced in the long run : con. 

 sequently no one can prudently play, even with the play 

 in his favour, unless he continue the occupation through 

 such a number of trials that he may reasonably expect 

 an average of all sorts of fortune. And though I have 

 hitherto appeared to speak only of games of chance, yet 

 precisely the same considerations apply to mercantile 

 speculations, and to every species of affair in which no 

 absolute certainty exists. If any possible event enter 

 into the play which, from the nature of the game, cannot 

 often occur, and if a stake be made upon the arrival of 

 that event, proportionate to some enormous benefit which 

 it is agreed that event shall secure, then prudence re- 

 quires that the game shall be very often repeated, or, if 

 that cannot be done, that it shall not be played at all. 

 There is a celebrated case known by the name of the 

 Petersburgh problem, which is one of the most in- 

 structive lessons in this subject, both on account of its 

 paradoxical appearance, and also because very eminent 

 writers have considered it as a sort of stumbling-block, 

 and have endeavoured to evade the conclusion. Con- 

 dor cet and others have taken a proper view of the subject; 

 while among those who have considered the problem as 

 an anomaly, we may instance D'Alembert. 



If p, g, and r be three fractions whose sum is unity, 

 it follows that we may suppose three events, one of which 

 must happen, and neither of the others, and of which the 

 chances are p, q, and r : and similarly of more fractions 

 than three, whose sum is unity ; or of any number of 

 fractions, however great, provided their sum be unity. 

 Let us, then, take an infinite number of fractions whose 

 sum is unity : either of the following series will do. 



